\(H_2\)-control and the separation principle for discrete-time Markovian jump linear systems
type: Item
instance of: scholarly article
title: \(H_2\)-control and the separation principle for discrete-time Markovian jump linear systems
zbMATH Open document ID: 1066.93065
author: E. F. Tuesta
author: Oswaldo Luiz V. Costa
published in: MCSS. Mathematics of Control, Signals, and Systems
publication date: 2005-02-08T00:00:00Z
review text: Let \((\Omega, \{ {\mathcal F}_k\}_{k \in T}, {\mathcal F}, P)\), \(T = \{ 0, 1, \ldots\}\) be a probabilistic space. The authors consider the following Markovian Jump Linear Systems \[ {\mathcal L}_G = \begin{cases} x(k+1) = A_{\theta(k)} x(k) + B_{\theta (k)} u(k) + G_{\theta(k)} w(k), \\ y(k) = L_{\theta (k)} x(k) + H_{\theta (k)} w(k), \\ z(k) = C_{\theta (k)} x(k) + D_{\theta (k)} u(k),\end{cases} \] where \(\{x(k), k \in T\}\) represents the state vector in \(C^n\), \(\{u(k), k \in T\}\) the control sequence in \(C^m\), \(\{w(k), k \in T\}\) the input sequence in \(C^q\), \(\{y(k), k \in T\}\) the sequence of measurable variables in \(C^p\), \(\{z(k), k \in T\}\) the output sequence in \(C^r\), and the output and ``operation models'' (\(y(k)\), \(\theta (k)\) respectively) are known at each time. A \(k\)-dynamic Markovian controller \(z_k\) for \({\mathcal L}_G\) is given by \[ {\mathcal L}_k = \begin{cases} {\hat x}(k+1) = {\hat A}_{\theta(k)} {\hat x}(k) + {\hat B}_{\theta (k)} y(k), \\ u(k) = {\hat C}_{\theta (k)} {\hat x}(k).\end{cases} \] The authors show that in the case with no jumps an optimal controller can be obtained from two sets of coupled algebraic Riccati equations, one associated with the optimal control problem when the state variable is available, and the other associated with the optimal filtering problem. This is the principle of separation for discrete-time Markovian jump linear systems. The authors consider an interesting example.
Mathematics Subject Classification ID: 93E20
Mathematics Subject Classification ID: 93C55
Mathematics Subject Classification ID: 60J75
zbMATH DE Number: 2132326
zbMATH Keywords: discrete-time system
zbMATH Keywords: \(H_2\)-control
zbMATH Keywords: infinite-horizon
zbMATH Keywords: Markovian jump systems
zbMATH Keywords: separation principle
zbMATH Keywords: coupled algebraic Riccati equations
zbMATH Keywords: optimal filtering
reviewed by: Yu. V. Kozachenko
MaRDI profile type: Publication
full work available at URL: s00498-004-0142-3
OpenAlex ID: W2040577692
DOI: 10.1007/S00498-004-0142-3
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description: scientific article; zbMATH DE number 2132326